On the existence and uniqueness of a strong solution to the antiperiodic problem for a 2-parabolic equation with a deviating argument

Abstract

This paper investigates the antiperiodic boundary value problem for a 2-parabolic equation with a time-deviating argument. A corresponding spectral problem is constructed, the symmetry of the differential operator is proven, and the properties of eigenvalues and eigenfunctions are established. It is shown that the eigenvalues have multiplicity two, and the corresponding eigenfunctions form an orthonormal basis in a Hilbert space. A strong solution to the problem is obtained in the form of a series expansion using the orthonormal basis of eigenfunctions corresponding to the spectrum of the operator generated by the boundary value problem. Conditions for the existence and uniqueness of a strong solution are established, and an explicit form of the inverse operator is constructed. Furthermore, it is proven that the problem operator is essentially self-adjoint. The proven statements complement the theoretical foundation for problems with deviating arguments in classes with antiperiodic boundary conditions, which is significant in modeling processes with memory and delay.